Fool’s Gold

They are close relatives, with all the same symmetries… but what excites me most is that they have 5-fold symmetry. It’s a theorem that no crystal can have 5-fold symmetry. So, we might wonder whether these shapes occur in nature… and if they don’t, how people dreamt up these shapes in the first place.

It’s widely believed that the Pythagoreans dreamt up the regular dodecahedron after seeing crystals of iron pyrite—the mineral also known as ‘fool’s gold’. Nobody has any proof of this. However, there were a lot of Pythagoreans in Sicily back around 500 BC, and also a lot of pyrite. And, it’s fairly common for pyrite to form crystals like this:

This crystal is a ‘pyritohedron’. It looks similar to regular dodecahedron—but it’s not! At the molecular level, iron pyrite has little crystal cells with cubic symmetry. But these cubes can form a pyritohedron:

(By the way, you can click on any of these pictures for more information.)

You’ll notice that the front face of this pyritohedron is like a staircase with steps that go up 2 cubes for each step forwards. In other words, it’s a staircase with slope 2. That’s the key to the math here! By definition, the pyritohedron has faces formed by planes at right angles to these 12 vectors:

On the other hand, a regular dodecahedron has faces formed by the planes at right angles to some very similar vectors, where the number 2 has been replaced by this number, called the golden ratio:

Namely, these vectors:

Since

the golden ratio is not terribly far from 2. So, the pyritohedron is a passable attempt at a regular dodecahedron. Perhaps it was even good enough to trick the Pythagoreans into inventing the real thing.

If so, we can say: fool’s gold made a fool’s golden ratio good enough to fool the Greeks.

At this point I can’t resist a digression. You get the Fibonacci numbers by starting with two 1’s and then generating a list of numbers where each is the sum of the previous two:

1, 1, 2, 3, 5, 8, 13, …

The ratios of consecutive Fibonacci numbers get closer and closer to the golden ratio. For example:

and so on. So, in theory, we could use these ratios to make cubical crystals that come closer and closer to a regular dodecahedron!

And in fact, pyrite doesn’t just form the 2/1 pyritohedron I showed you earlier. Sometimes it forms a 3/2 pyritohedron! This is noticeably better. The 2/1 version looks like this:

while the 3/2 version looks like this:

Has anyone ever seen a 5/3 pyritohedron? That would be even better. It would be quite hard to distinguish by eye from a true regular dodecahedron. Unfortunately, I don’t think iron pyrite forms such subtle crystals.

Okay. End of digression. But there’s another trick we can play!

These 12 vectors I mentioned:

besides being at right angles to the faces of the dodecahedron, are also the corners of the icosahedron!

And if we use the number 2 here instead of the number , we get the vertices of a so-called pseudoicosahedron. Again, this can be made out of cubes:

However, nobody seems to think the Greeks ever saw a crystal shaped like a pseudoicosahedron! The icosahedron is first mentioned in Book XIII of Euclid’s Elements, which speaks of:

the five so-called Platonic figures which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and the icosahedron are due to Theaetetus.

So, maybe Theaetetus discovered the icosahedron. Indeed, Benno Artmann has argued that this shape was the first mathematical object that was a pure creation of human thought, not inspired by anything people saw!

That idea is controversial. It leads to some fascinating puzzles, like: did the Scots make stone balls shaped like Platonic solids back in 2000 BC? For more on these puzzles, try this:

But right now I want to head in another direction. It turns out iron pyrite can form a crystal shaped like a pseudoicosahedron! And as Johan Kjellman pointed out to me, one of these crystals was recently auctioned off… for only 47 dollars!

Most viruses with 5-fold symmetry have protein shells in patterns based on the same math as geodesic domes:

But some more unusual viruses, like polyomavirus and simian virus 40, use more sophisticated patterns made of two kinds of tiles:

They still have 5-fold symmetry, but these patterns are spherical versions of Penrose tilings! A Penrose tiling is a nonrepeating pattern, typically with approximate 5-fold symmetry, made out of two kinds of tiles:

To understand these more unusual viruses, Twarock needed to use some very clever math:

I am not a physics graduate but I have a big interest in physics. Recently I was reading some Group Theory for Physics texts and as it is usual the text starts with crystallographic groups. Unfortunately the analysis of the said text was not good or thorough. Can you recommend a text(s) for crystallographic groups with all the hard core analysis? Classification, defintions and so on.

Dear Vasileios – I don’t really have a favorite book on this subject. Maybe someone else can recommend one. And maybe you can tell use which books you were reading, so we can know what you consider insufficient.

Back home in California I have Group Theory and Its Application to Physical Problems by Morton Hamermesh. This has a lot of material on crystallographic groups. But I never really loved that book.

I am very fond of Conway, Doyle, Gilman and Thurston’s classification of wallpaper groups using orbifolds, which I discussed in "week267" of This Week’s Finds. Wallpaper groups are the 2d analogue of crystallographic groups.

Conway, Friedrichs, Huson, and Thurston have generalized this orbifold argument to crystallographic groups. But that’s probably not what you want! Their work seems to assume we already know the statement of the result to be proved.

First chemists speculated that carbon might form C60 molecules shaped like soccer balls, and later they managed to produce them. In took a bit of work to get ahold of at first… but later they realized that lowly soot contains lots of this stuff!

Since Buckminster Fuller was fond of this shape in his geodesic domes—with each pentagon or hexagon filled in by triangles—C60 and its relatives are called fullerenes, and the shape is affectionately called a buckyball. For more about this stuff, try:

Mathematically, we get the shape of a buckyball by
taking an icosahedron and chop off all 12 corners. The result is called a ‘truncated icosahedron’. It has 12 regular pentagonal faces, one for each corner of the original icosahedron. It has 20 regular hexagonal faces, one for each face of the original icosahedron. All its edges are the same length.

It’s called an Archimedean solid because, while not quite Platonic in its beauty, every face is a regular polygon and every vertex looks alike: two pentagons abutting one hexagon.

And now, some more fancy stuff!

Here’s a story about icosahedra, buckyballs, and last letter Galois wrote before his famous duel, taken from "week79" of This Week’s Finds, but based on this marvelous paper:

When I was a graduate student at MIT I realized that Kostant (who teaches there) was deeply in love with symmetry, and deeply knowledgeable about some of its more mysterious byways. Unfortunately I didn’t dig too deeply into group theory at the time, so I spent years later trying to catch up.

Let’s start with the Platonic solids. Note that the cube and the octahedron are dual—putting a vertex in the center of each of the cube’s faces gives you an octahedron, and vice versa. So every rotational symmetry of the cube can be reinterpreted as a symmetry of the octahedron, and vice versa. Similarly, the dodecahedron and the icosahedron are dual, while the tetrahedron is self-dual. So while there are 5 Platonic solids, there are really only 3 different symmetry groups here.

These 3 "Platonic groups" are very interesting. The symmetry group of the tetrahedron is the group A4 of all even permutations of 4 things, since by rotating the tetrahedron we can achieve any even permutation of its 4 vertices. The symmetry group of the cube is S4, the group of all permutations of 4 things. What are the 4 things here? Well, we can draw 4 line segments connecting opposite vertices of the cube; these are the 4 things! The symmetry group of the icosahedron is A5, the group of even permutations of 5 things. What are the 5 things? It we take all the line segments connecting opposite vertices we get 6 things, not 5, but we can’t get all even permutations of those by rotating the icosahedron.

To find the 5 things is a bit trickier; I leave it as a puzzle here. See:

Once we convince ourselves that the rotational symmetry group of the icosahedron is A5, it follows that it has 5!/2 = 60 elements. But there is a much easier way to see this.

The truncated icosahedron has 5 × 12 = 60 vertices. Every symmetry of the icosahedron is a symmetry of the truncated icosahedron, so A5 acts to permute these 60 vertices. Moreover, we can find an element of A5 that moves a given vertex of the truncated icosahedron to any other one, since "every vertex looks alike". Also, there is a unique element of A5 that does the job. So there must be precisely as many elements of A5 as there are vertices of the truncated icosahedron, namely 60.

Kostant’s paper requires more knowledge of group theory to understand. He focuses on a fact buried in Galois’ last letter, written to the mathematician Chevalier on the night before Galois’ fatal duel. He was thinking about some groups we’d now call PSL(2,F). Here F is a field (for example, the real numbers, the complex numbers, or Zp, the integers mod p where p is prime). PSL(2,F) is a "projective special linear group over F."

What does that mean? Well, first of all, SL(2,F) is the 2×2 matrices with entries in F having determinant equal to 1. These form a group under good old matrix multiplication. The matrices in SL(2,F) that are scalar multiples of the identity matrix form the "center" Z of SL(2,F)—the group of guys who commute with everyone else. We can form the quotient group SL(2,F)/Z, and get a new group called PSL(2,F).

Now Galois was thinking about PSL(2,Zp) where p is prime. There’s an obvious way to get this group to act as permutations of p+1 things. Here’s how! For any field F, the group SL(2,F) acts as linear transformations of the 2-dimensional vector space over F, and it thus acts on the set of lines through the origin in this vector space… which is called the "projective line" over F. But anything in SL(2,F) that’s a scalar multiple of the identity doesn’t move lines around, so we can mod out by the center and think of the quotient group PSL(2,F) as acting on projective line. (By the way, this explains the point of working with PSL instead of plain old SL.)

Now, an element of the projective line is just a line through the origin in F2. We can specify such a line by taking any nonzero vector (x,y) in F2 and drawing the line through the origin and this vector. However, (x’,y’) and (x,y) determine the same line if (x’,y’) is a scalar multiple of (x,y). Thus lines are in 1-1 correspondence with vectors of the form (1,y) or (x,1). When our field F is Zp, there are just p+1 of these. So PSL(2,Zp) acts naturally on a set of p+1 things.

What Galois told Chevalier is that PSL(2,Zp) doesn’t act nontrivially as permutation of any set with fewer than p+1 elements if p > 11. This presumably means he knew that PSL(2,Zp) does act nontrivially on a set with only p elements if p = 5,7, or 11. For example, PSL(2,5) turns out to be isomorphic to A5, which acts on a set of 5 elements in an obvious way. PSL(2,7) and PSL(2,11) act on a 7-element set and an 11-element set, respectively, in sneaky ways which Kostant describes.

These cases, p = 5, 7 and 11, are the the only cases where this happens and PSL(2,Zp) is "simple" in the technical sense of group theorists. In each case it is very amusing to look at how PSL(2,Zp) acts nontrivially on a set with p elements and consider the subgroup that doesn’t move a particular element of this set. For example, when p = 5 we have PSL(2,5) = A5, and if we look at the subgroup of even permutations of 5 things that leaves a particular thing alone, we get A4. Kostant explains how if we play this game with PSL(2,7) we get S4, and if we play this game with PSL(2,11) we get A5. These are the 3 Platonic groups again!!

But notice an extra curious coincidence. A5 is both PSL(2,5) and the subgroup of PSL(2,11) that fixes a point of an 11-element set. This gives a lot of relationships between A5, PSL(2,5), and PSL(2,11). What Kostant does is take this and milk it for all it’s worth! In particular, it turns out that one can think of A5 as the vertices of the buckyball, and describe which vertices are connected by an edge using the embedding of A5 in PSL(2,11). I won’t say how this goes… read his paper!

Keef and Twarock’s is a fascinating paper, thanks for passing it on. Although I am not up to otherwise comment on it, it prompts me to document a fun fact that I haven’t seen documented anywhere, and that has at least in common with their work to call for looking into 4D analogues (in turn a fun idea by itself, that mathematics would now allow to dream up 4D viruses in non-trivial logical detail).

I believe the right concise wording for my fun fact – and I would welcome rectification – is to say that the surface of the regular platonic dodecahedron
is isometric to the outer surface or shell of both Poinsot’s great dodecahedron
and to that of the first stellation of the icosahedron (aka small triambic icosahedron).
All three have icosahedral symmetry. The latter two are self-intersecting polyhedra so that a distinction needs to be made between their surface and the part of it that’s apparent to the outside; what I call the outer shell and what is usually the only surface considered when making paper models. To restate my fun fact in simpler words, if you imagine these surfaces made out of paper, it is in principle possible to refold any one of the three without any cutting, to obtain any of the other two. Easier in practice would be to cut up the surface enough to lay it out flat and then refold it and glue together the lips of the cuts.

This property has been well known to paper modelists for a long time in the case of the two self-intersecting polyhedra, but AFAIK it hadn’t yet been remarked that both are similarly related in turn to the regular dodecahedron. The reason this is not obvious is that the mapping from or to the regular dodecahedron preserves no edge or face, while in the case of the other two polyhedra the mapping relates both faces and edges one-to-one (when considering the outer shells as polyhedra in their own right, or else we probably need to speak of “folds” for edges and “flat polygons” for faces).

On the left is a flat surface from which either one of the three polyhedra displayed can be constructed. A single pentagon from the regular dodecahedron is shown in red, while in green is shown a single triangular face of the outer shell of either a great dodecahedron or a small triambic icosahedron.

The property I expose here has been well known to paper modelists for a long time in the case of the two self-intersecting polyhedra, but AFAIK the remark that both are similarly related to the regular dodecahedron, is new. The reason this was not obvious is that starting from the regular dodecahedron the mapping preserves no edge or face, while in the case of the self-intersectors the relationship relates the faces and edges one-to-one (when considering the outer shells as polyhedra in their own right, or else we probably need to speak of “folds” for edges and “flat polygons” for faces).

Nice stuff, Boris! I’m sorry the wonderful images in your comment did not appear at first. Because this is a free WordPress blog, it’s impossible for anyone but me to include images in their comments, and it’s also impossible for people to preview their comments. Sorry!

However, I routinely fix mistakes in people’s comments. And
if someone wants to include an image, they can just include the URL, e.g.

and I’ll do the rest. (Of course, this takes me a bit of work, so I’d appreciate it if people include images sparingly.)

To me the most interesting thing about Keef and Twarock’s work is that it suggests a relation between the more sophisticated patterns for viral protein coats and the higher-dimensional geometry underlying quasiperiodic tilings and quasicrystals.

For a great introduction to the higher-dimensional geometry behind quasiperiodic tilings, see:

Johan Kjelmann, who works in the mineral collection at Uppsala Universitet, recently found some interesting crystals made of cobaltite (CoAsS). They’re from Tunaberg, Sweden. They’re quite small, but they’re shaped like pseudoicosahedra!

Johan notes that you can see more about cobaltite, including a rotating picture of a pseudoicosahedron, here:

Johan Hjellman sent me an email that’s too interesting to keep to myself. So, I’ll post it here with a few small improvements. The most interesting part to me is that the pyrite crystals forming pyritohedra of type 5/3 and 13/8 have also been seen! The latter would be visually indistinguishable from a regular dodecahedron.

Hi, I browsed your blog. Interesting and informative stuff.

First comment is that pyrite crystals are not built up of small cubic cells, but the symmetry of the “building block”—the unit cell—is cubic, Alas, it is OK as an image. Thus, your sentence “At the molecular level, iron pyrite has little cubic crystal cells” would be better as follows: “At a molecular level, iron pyrite has a unit cell with (of?) cubic symmetry” or similar.

The concept that crystals are built up of polyhedra was introduced by the french mineralogist Rene Just Haüy around 1800 and his images of stacked crystal models became standard in the crystallography parts of the mineralogy books of the 19th c. I was quite surprised and happy to see them on your site in a modern context. Haüy was also heading a small manufacture of real wooden models that were sold from Paris in the first decades of the 19th c. We have a couple in our collection; my attachment shows “structure de dodecahedre pentagonal”, but there are some great collections in Berlin, Vienna, etc comprising several hundred models.

Second comment, as I am not a mathematician, it’s all a bit tricky—anyhow one concept I understood is your discussion on the pentagonal dodecahedra and their relation to Fibonacci numbers. I understand that the numbers are getting closer to the golden ratio, and you mentioned the 3/2 pyritohedron that is even a better approximation to the regular dodecahedron. So I went to a mineralogical classic- Hinte’s Handbuch der Mineralogie and checked out what Miller indices have actually been observed on pyrite and it turns out that the forms (5.3.0) and (13.8.0) have also been observed.

John, I haven’t read the original references for the forms (5.3.0) and (13.8.0) – for sure if any of these forms were the only one present on the crystals they would for sure be fantastic near perfect regular dodecahedra – but most likely they only appeared as tiny faces modifying crystals of more common forms.

By the way, talking as a devils advocate, I admire Twarocks virus tiling, but did anyone calculate the actual (spherical) angles of the tiles on the sphere? They cannot be very Penrose-tile-like, since such a tiling would be mostly planar, given its connectivity as shown for the virus. Thus, they have to be distorted? And another thought: Twarock combines kites and large rhombs, taking tiles from both distinct Penrose tilings that would not match in the plane, but what about the other choice, darts and small rhombs, would they tile a spherical surface, too? I wonder why mathematicians most often still do not look at “less-ideal” cases, there would be an awful amount of discoveries waiting.

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